For any positive integer $n$, consider
$$\int_{0}^\infty\frac{(r^2-1)r^{n+1}}{(r^2+1)^{n+3}}dr.$$
I would like to show that it is positive. I try to write it as
$$\int_{0}^\infty\frac{(r^2-1)r^{n+1}}{(r^2+1)^{n+3}}dr=\int_{0}^\infty\frac{r^{n+1}}{(r^2+1)^{n+2}}dr-2\int_{0}^\infty\frac{r^{n+1}}{(r^2+1)^{n+3}}dr,$$
but I am not sure that it helps.

EDIT: According to sjasonw, the integral may not be positive as I think. I would be happy to see a proof showing that it's not positive.